5 Noise Reduction in Amplitude-Fluctuation Electronic Speckle-Pattern Interferometry 31 Fig. 5.4 Sample fringe images (a) high contrast and (b) low contrast 5.2.4 Introduction of Carrier Fringe System Now we discuss the carrier fringe configuration (the case of the tilted specimen indicated with the dashed line in Fig. 5.3). In this case the factor cosıo can be replaced by the spatial variation as follows. S. / Š2I0 f1Ccos˛xJ0 .ı/g (5.7) Unlike the normal incident configuration, the operation-point phase varies through multiple periods in a given image; a dark fringe corresponds to cosıo D /2CN and the neighboring bright fringe to cosıo DN or cosıo D(NC1) (N: integer). The fluctuation of ıo, whatever its cause many be, changes the position of dark fringes or shifts the x coordinate points that correspond to the condition cosıo D /2CN . In the space domain, the fringe shifts reduce the contrast. This indicates the possibility to correlate the fringe contrast reduction to the oscillation amplitude ı. However, in the space domain, the correlation is not easy to quantify. Figure 5.4 illustrates typical fringe images of high and low contrasts. Instead, in the spatial frequency domain, it is easily quantified. We will discuss this in the next sections. It should be noted that low frequency phase noise causes the carrier fringes to shift in the space domain but as long as the angle of tilt made by the specimen remains the same the fringe spacing is intact. Therefore, in the frequency domain the peak frequency is unaffected by low frequency phase noise. 5.3 Experimental Results and Discussions 5.3.1 Normal Incident Configuration Figure 5.5 shows sample reference and signal beam intensity profiles of a two-dimensional phase analysis with the nominal incident configuration. It is seen that while the reference beam profile is Gaussian, the profile of the signal beam is deformed. The reason for the deformed profile is most likely that the specimen surface is not flat at the level of the laser wavelength and the reflectivity is not uniform. Figure 5.6 shows the interference terms calculated by subtracting the sum of the reference and signal beam intensities (Fig. 5.5) from the signal intensity on a pixel-by-pixel basis. The reference beam and signal beam intensities were obtained by blocking one arm at a time. According to Eq. (5.5), the resultant quantity can be expressed as follows. S. / 2I0 2I0 D cosıoJ0 .ı/ (5.8) The two interference intensities were taken at two times steps 10 frames or approximately 300 ms apart. (Since the frame rate was 30 fps (frames per second), the separation in time for 10 frames is 10/30 D0.333 s.) During this measurement, the acoustic driving frequency (hence J0(ı)) was unchanged. Therefore the difference between the two intensity profiles in Fig. 5.6 is due to the change in the phase ıo in the cosıo term. The following observations can be made in regard to the two interference intensities. First they both are centered at cosıoJ0(ı) D0.5. This indicates that cosıo happens to be 0.5, or ıo Dcos 1(0.5) D1.05(rad). Note that J 0(0)D1 and those regions where the effect of the oscillation is negligible (i.e., the region far from the center of the beam) cosıoJ0(ı) Dcosıo. Second, the intensity profile at frame # 2 bulges more on the positive side whereas the one at frame # 12 bulges more on
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